PYTHAGOREAN Theorem & Triples

Simple Geometric Proof of Pyth. Theorem. Animation without words

Why 3,4,5? Why 3 by 4 right triangle has hypotenuse 5 - EXACTLY? Does this make sense?

Wrong Legs. Start with any two "wrong" legs and end up with an EXACT Pyth. Triple

Pyth TRIPLES. Shows all possible triples with a given side-length. Enter a Leg in the window "Evn+2" if it is even, or in the window "Odd+2" if it is odd, and press "Enter" key. Further clicking on the corresponding button will consecutively INCREASE THAT LEG by 2 and show all possible triples. You can always enter a Leg and press "Enter" key. Or click "Restart" button to start all over. "Enter" the "Hypotenuse" in its window, and click "Hyp+1" button to check them consecutively. In all cases, multiple solutions come with bigger numbers.

Here is a GAME you can play using this applet. Start from any triple (or click the buttons randomly to mess up). The goal is: By clicking the Buttons ONLY to arrive to the smallest possible triple? What is it? Can it be 3,4,5? Can it be 5,12,13? ... Can you arrive to not the smallest but any other given triple?

Legs' Difference. Creating Pyth. Triples with any given difference of Legs. That difference starts with 1 by default, but clicking button "Difference +" you can increase it by 1. Clicking the "Go Higher" takes up to a higher values (above 1000). Watch he Status bar of your internet browser.


SOME FACTS about Pythagorean Triples

Verify that for any m and n: (m-square + n-square)-squared minus (m-square - n-square)-squared equals (2mn)-squared.

For any Pythagorean triple, not only the square of the hypotenuse is a sum of two squares (of the leg-numbers), but the hypotenuse itself is a sum of squares (of other two numbers).

Examples: 3,4,5. Not only 5 is square root of 25, which is 3-squared plus 4-squared, but 5 itself also is 1-squred plus 2-squared!

Example: 5,12,13. Not only 13 is a square root 169 which is 5-squared plus 12-squared, but 13 itself is also 3-squared plus 2-squared.

Every prime (but 2) is a LEG of some Pythagorean Triple.

Generally, any whole number (greater than 2) is a leg of some Pythagorean Triple.

The difference between Hypotenuse and a Leg is a square or twice a square (of any whole number)! The difference between the Legs can be any whole number!

Statistically, one out of four whole numbers is a Hypotenuse of some Pythagorean Triple.